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Optics Express

  • Editor: J. H. Eberly
  • Vol. 3, Iss. 5 — Aug. 31, 1998
  • pp: 171–179
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Vacuum-induced jitter in spatial solitons

Elna M. Nagasako, Robert W. Boyd, and G. S. Agarwal  »View Author Affiliations


Optics Express, Vol. 3, Issue 5, pp. 171-179 (1998)
http://dx.doi.org/10.1364/OE.3.000171


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Abstract

We perform a calculation to determine how quantum mechanical fluctuations influence the propagation of a spatial soliton through a nonlinear material. To do so, we derive equations of motion for the linearized operators describing the deviation of the soliton position and transverse momentum from those of a corresponding classical solution to the nonlinear wave equation, and from these equations we determine the quantum uncertainty in the soliton position and transverse momentum. We find that under realistic laboratory conditions the quantum uncertainty in position is several orders of magnitude smaller the classical width of the soliton. This result suggests that the reliability of photonic devices based on spatial solitons is not compromised by quantum fluctuations.

© Optical Society of America

1. Introduction

Spatial solitons are beams of light that propagate with a constant transverse dimensions as the result of an exact balance between diffraction and self focusing effects [1

1. A pedagogical discussion of self-action effects including self-trapping and optical solitons is presented in Chapter 6 of R. W. Boyd, Nonlinear Optics (Academic, San Diego, 1992). See also E. M. Nagasako and R. W. Boyd, in Amazing Light, A volume dedicated to Charles Hard Townes on his 80th Birthday, edited by R. Y. Chiao (Springer, New York, 1996).

]. The possibility of such an occurrence was recognized in some of the very early studies of nonlinear optical phenomena [2

2. G. A. Askar’yan, Sov. Phys. JETP15, 1088 (1962).

,3

3. R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett.13, 479 (1964). [CrossRef]

], and the relation between spatial solitons and temporal solitons was elucidated by Zakharov and Shabat [4

4. V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP34, 62 (1972).

]. Although the balance between diffraction and self focusing can occur in either one or two transverse dimensions, the equilibrium condition is unstable in the two-dimensional case for a material with a purely third order nonlinear susceptibility [5

5. E. L. Dawes and J. H. Marburger, Phys. Rev.179, 862 (1969); See also J. H. Marburger, Prog. Quantum Electron.4, 35 (1975). [CrossRef]

,6

6. Y. Silberberg, Opt. Lett.15, 1282 (1990). [CrossRef] [PubMed]

]. Spatial solitons, however, are predicted to be stable and have been observed experimentally in one transverse dimension [7–9

7. J. S. Aitchison, Y. Silberberg, A. M. Weiner, D. E. Leaird, M. K. Oliver, J. L. Jackel, E. M. Vogel, and P. W. E. SmithJ. Opt. Soc. Am. B8, 1290 (1991). [CrossRef]

], as occurs for instance in the case of propagation through a planar waveguide.

Numerous technological applications based on the properties of spatial solitons have been proposed, including optical interconnects [10

10. B. Luther-Davies and X. Yang, Opt. Lett.17, 1755 (1992). [CrossRef] [PubMed]

], optical logic gates [11

11. S. Blair, K. Wagner, and R. McLeod, Opt. Lett.19, 1943 (1994). [CrossRef] [PubMed]

], and other optical switching devices. The operational integrity of any such device requires that the spatial jitter in the position of the spatial soliton be held to a minimum. In the present paper, we describe a theoretical investigation of the influence of quantum noise on the propagation of spatial solitons, and in particular derive expressions that allow us to determine the uncertainty in the position of a spatial soliton after propagating through a nonlinear optical medium. As described below, we find that quantum fluctuations can lead to a small but measurable uncertainty in the soliton position. We earlier studied the influence of quantum fluctuations on the initiation of the filamenta-tion process [12

12. E. M. Nagasako, R. W. Boyd, and G. S. Agarwal, Phys. Rev. A55, 1412 (1997). [CrossRef]

].

There is a considerable body of prior work involving the influence of quantum noise on the properties of temporal solitons. Much of this work was motivated by issues associated with the reliability of optical fiber telecommunications systems. For instance, Gordon and Haus [13

13. J. D. Gordon and H. A. Haus, Opt. Lett.11, 665 (1986). [CrossRef] [PubMed]

] have considered the influence of noise associated with the amplification of solitons signal trains in a long-distance telecommunications system. The influence of quantum fluctuations on propagation of solitons through a passive optical fiber system has also been considered, for instance by Drummond and co-workers [14

14. P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamomoto, Nature, 365, 307 (1993). [CrossRef]

] and by Haus and co-workers [15

15. H. A. Haus and M. N. Islam, IEEE J. Quantum Electron.QE-21, 1172 (1985). [CrossRef]

,16

16. H. A. Haus and Y. Lai, J. Opt. Soc. Am. B7, 386 (1990). [CrossRef]

]. The calculation presented in the present paper follows much of the same methodology as that used by Haus and Lai [16

16. H. A. Haus and Y. Lai, J. Opt. Soc. Am. B7, 386 (1990). [CrossRef]

] in their treatment of temporal solitons. Our work differs from theirs in that we use the formalism to treat spatial rather than temporal solitons. In addition, we are particularly interested in determining the uncertainty in the transverse soliton position, whereas Haus and Lai used their formalism to treat quadrature squeezing in the transmitted soliton. In addition, we establish some mathematical relations such as the definition of the adjoint operation in the context of the present problem (Appendix A) and we establish the sense in which soliton propagation can be described by the same equations as the quantum mechanical treatment of a free particle (Appendix B).

2. Theoretical Formulation

Our theoretical development starts with the nonlinear Schrodinger equation with variation in one transverse spatial dimension

2ikzA+2x2A+2k2n̄2n0A2A=0,
(1)

where the amplitude A is defined according to the convention

E(r,t)=12x̂[A(x,z)exp(iβ0ziω0t)+c.c.].
(2)

We make the change of variables

X2kxZkzc1/2
Φ(+)An0/n̄2Φ()(Φ(+))*A*n0/n̄2

where 2 = (n 0 c/4π)n 2 where n 2 is the usual nonlinear refractive index defined such that Δn = n 2 I. We then find that the nonlinear Schrodinger equation (for the positive frequency part of the field) takes the standard [16

16. H. A. Haus and Y. Lai, J. Opt. Soc. Am. B7, 386 (1990). [CrossRef]

] normalized form [17

17. Although c has the fixed value -1/2 using the present conventions, we retain c in our formulas for more ready comparisons of our results with those obtained using different normalization conventions.

]

iZΦ(+)+2X2Φ(+)2cΦ(+)2Φ(+)=0.
(3)

The fundamental soliton solution to this equation is given by

Φ0(+)(X,Z)=N0c1/22exp[iN02c24Zip02Z+ip0(XX0)+iθ0]
sech[N0c2(XX02p0Z)],
(4)

where the quantity

N0=Φ(+)(X,Z)2dX

represents the soliton intensity profile integrated over the transverse spatial dimension X, θ 0 is the soliton phase, p 0 is the transverse soliton momentum, and X 0 is the position of the soliton center.

Φ̂(+)(X,Z)=Φ(+)(X,Z)+Ψ̂(+)(X,Z).
(5)

We also introduce the negative frequency part of the field operators

Ψ̂()(X,Z)=Ψ̂(+)(X,Z),Φ̂()(X,Z)=Φ̂(+)(X,Z),

etc. The perturbation is assumed to obey the quantum mechanical commutation relation

[Ψ̂(+)(X,Z),Ψ̂()(X,Z)]=Sδ(XX)S=ħck02n22ΔtLy.
(6)

where Δt is the response time of the nonlinear response and Ly is the thickness of the slab waveguide that confines the radiation in the y direction. The form of the coefficient S is obtained by requiring that the total electric field obey the standard field commutation relation and then reducing the problem to one transverse dimension and one frequency component, under the assumptions that the spectrum is essentially uniform over the frequency interval 1/Δt and that the field amplitude is essentially constant over the thickness Ly of the waveguide. By substituting expression (5) into the nonlinear Schrodinger equation (1) and linearizing in the perturbation, the evolution equation for the perturbation

iZΨ̂(+)+2X2Ψ̂(+)+4cΦ0(+)2Ψ̂(+)+2c(Φ0(+))2Ψ̂()=0
(7)

is obtained. It has been shown previously [16

16. H. A. Haus and Y. Lai, J. Opt. Soc. Am. B7, 386 (1990). [CrossRef]

] that to a good approximation the perturbation ψ^ (+) can be represented as the sum of contributions resulting from fluctuations in the four fundamental soliton degrees of freedom:

Ψ̂(+)(X,Z)=Φ0(+)N0ΔN̂0(+)+Φ0(+)θ0Δθ̂0(+)+Φ0(+)p0Δp̂0(+)+Φ0(+)X0ΔX̂0(+).
(8)

In writing this equation in the form shown, we have ignored the continuum contribution to the perturbation. The derivatives which form the coefficients of the fluctuation operators in this expression are also solutions of Eq. (7). These solutions can be denoted by

Ψj(+)(X,Z)=Φ0(+)(X,Z)j0,
(9)

where j = N, θ, p and X and are given explicitly by

ΨN(+)(X,Z)=[1N0+iN0c22Zc2Xtanh(N0c22X)]Φ0(+)(X,Z),
Ψθ(+)(X,Z)=iΦ0(+)(X,Z),
Ψp(+)(X,Z)=[iX+N0cZtanh(N0c2X)]Φ0(+)(X,Z),
ΨX(+)(X,Z)=[N0c2tanh(N0c2X)]Φ0(+)(X,Z).
(10)

Using the derivative solutions ψj(+) (X,Z) as well as their adjoints ψ_ j (+)(X,Z) (see Appendix A for a description of the adjoint operation in the present context), we can now invert the expansion of the perturbation operator ψ^ (+) to give the individual fluctuation operators as follows :

ΔN̂0=[Ψ̲θ()(X,0)Ψ̂(+)(X,0)+Ψ̲θ(+)(X,0)Ψ̂()(X,0)]dX,
Δθ̂0=[Ψ̲N()(X,0)Ψ̂(+)(X,0)+Ψ̲N(+)(X,0)Ψ̂()(X,0)]dX,
Δp̂0=1N0[Ψ̲X()(X,0)Ψ̂(+)(X,0)+Ψ̲X(+)(X,0)Ψ̂()(X,0)]dX,
ΔX̂0=1N0[Ψ̲p()(X,0)Ψ̂(+)(X,0)+Ψ̲p(+)(X,0)Ψ̂()(X,0)]dX.
(11)

Here

ΔN̂0=ΔN̂0(+)+ΔN̂0()=ΔN̂0(+)+H.c.

represents the complete (i.e. Hermitian) fluctuation operator, and ΔN^0(+) its positive frequency part, and so on. Eqs. (11) can be verified by direct calculation making use of Eqs. (8) and (10). Note that these are the fluctuation operators relevant to the entrance plane to the material. Using these expressions, the following commutation relationships involving the fluctuation operators can be calculated:

[ΔN̂0,Δθ̂0]=iS
[ΔX̂0,N0Δp̂0]=iS.
(12)

These commutation relations are consistent with the view that the pair Δ0 and Δθ̂0 and the pair Δ0 and N 0Δ0 form conjugate operator pairs, and lead to uncertainty relations between the conjugate quantities. We shall see below (Eq. (22)) that the parameter N 0 is typically smaller than unity, with the consequence that uncertainty relation expressed in terms of position Δ0 and momentum Δ0 (without the factor of N 0) predicts a quantum mechanical uncertainty increased over the single particle case by a factor of 1/N 0.

The fluctuation operators at any position Z within the interaction region can be determined similarly. For simplicity, we consider only the case of a soliton propagating along the Z axis, that is, we set p 0 and X 0 equal to zero. We then obtain the results

ΔN̂(Z)=[Ψ̲θ()(X,0)exp(iN02c2Z4)Ψ̂(+)(X,Z)
+Ψ̲θ(+)(X,0)exp(iN02c2Z4)Ψ̂()(X,Z)]dX,
Δθ̂(Z)=[Ψ̲N()(X,0)exp(iN02c2Z4)Ψ̂(+)(X,Z)
+Ψ̲N(+)(X,0)exp(iN02c2Z4)Ψ̂()(X,Z)]dX,
Δp̂(Z)=1N0[Ψ̲X()(X,0)
exp(iN02c2Z4)Ψ̂(+)(X,Z)
+Ψ̲X(+)(X,0)exp(iN02c2Z4)Ψ̂()(X,Z)]dX,
(13)
ΔX̂(Z)=1N0[Ψ̲p()(X,0)exp(iN02c2Z4)Ψ̂(+)(X,Z)
+Ψ̲p(+)(X,0)exp(iN02c2Z4)Ψ̂()(X,Z)]dX.

We mentioned above in connection with Eqs. (12) that Δ0 and N 0Δ0 obey quantum mechanical commutation relations. Moreover, we show in Appendix B that the soliton position fluctuation operator Δ and momentum fluctuation operator Δ obey the equations

ddZΔp̂=0
ddZΔX̂=2Δp̂,
(14)

which are the quantum mechanical equations of motion for a free particle with the formal substitution Zt. The factor of 2 in the second of Eqs. (14) results as a consequence of the particular conventions used in this calculation. These equations can be integrated to express the output fluctuations in terms of the input fluctuations as follows

Δp̂(Z)=Δp̂0
ΔX̂(Z)=ΔX̂0+2Δp̂0Z.
(15)

We can now use these results to determine the quantum mechanical uncertainty in the position of the soliton and to determine how this uncertainty changes with propagation distance Z. As before, we consider the case of a soliton propagating along the Z axis, that is, we set p 0 and X 0 equal to zero. Then through use of Eqs. (11) we find that the initial uncertainties in the soliton position and momentum, that is, the uncertainties at the entrance plane (Z = 0) are given by

ΔX̂02=π2S3N03c2
(16)

and

Δp̂02=N0c2S12.
(17)

We can then use the result of the second of the Eqs. (15), to find that the uncertainty at some arbitrary point Z in the material is given by

ΔX2(Z)=π2S3N03c2+N0c2S3Z2.
(18)

A(z)=Apeaksech(xw)exp(iz2kw2)
(19)

where

Apeak=1kwn02n̄2.
(20)

Note that the maximum change in refractive index

Δn=12n̄2Apeak2

is given by

Δn=n02k2w2.
(21)

By comparison of Eq. (19) with Eq. (4), with p 0, X 0, and θ 0 set equal to zero, we find that the parameter N 0 of the normalized solution is given by

N0=2kw.
(22)

Through use of Eqs. (18) through (22), and the variable change used in writing Eq. (3), we finally find that the mean squared position uncertainty in physical units can be expressed as

Δx2(z)=π2S(kw)w212+S12(kw)z2.
(23)

3. Discussion and Conclusions

Eq. (23) presents the primary result of the present calculation. We recall that this calculation is based on a linearization assumption, and thus strictly speaking its limits of validity require that the second term on the right hand side of this equation be much smaller than the first. Thus Eq. (23) is accurate for propagation distances that satisfy the inequality zπkw 2/2. For any value of z that satisfies this condition, including z = 0, the uncertainty in position is dominated by the first term. Under these conditions, the fractional uncertainty in soliton position will be given approximately as

(Δx)rmswπ212Skw.

We can evaluate this expression under typical laboratory conditions. We asssume that that k 0 = 1.3 × 105cm-1, corresponding to a wavelength of 0.5 μm, and that w and Ly are both of the order of 50 μm. Material parameters enter into the calculation as the ratio n 2t. This ratio tends to be nearly constant for most nonlinear optical materials, because materials that display a large nonlinear respone tend to be slow. One of the largest values of this ratio occurs for the conjugated polymers, for which n 2 can be as large as 3×10-12cm2/W and for which the response time Δt is believed to be as short as 10 fs. We then find that (Δx)rms/w ≈ 10-3, which is at best barely measurable in the laboratory. Of course, the predicted fractional uncertainty in soliton position would be larger if calculated for a material with a larger value of n 2t. Note also that Eq. (23), if extrapolated to values of z outside of its demonstrated limits of validity, predicts a linear increase in the fractional uncertainly in soliton position with increasing propagation distance z. It is not clear at present what the accurately predicted uncertainty would be under these conditions.

In summary, we have presented an analysis of the influence of quantum mechanical zero-point fluctuations on the propagation of spatial solitons, and have found that these fluctuations can lead to an uncertainty in the transverse position of the soliton. For most realistic conditions with currently available materials, this predicted fractional uncertainty is at best one part in 103 and consequently is barely experimentally observable. However, the fractional uncertainty could be considerably larger through use of materials with a larger nonlinear respponse or under conditions that lie outside of the validity of the present theory. These larger fractional encertainties in soliton position could have considerable importance for the construction of optical switching devices that rely on the properties of spatial solitons.

4. Acknowledgments

This work was supported by NSF grants INT 9712760 and ECS 9223726, by a US Army URI award, and by the sponsors of the University of Rochester’s Center for Electronic Imaging Systems. We thank G-L Oppo for converting this manuscript into LaTex, and we thank an unknown referee for insightful comments regarding our manuscript.

Appendix A

In this appendix, we present a discussion of the concept of the adjoint to the classical solution of the linearized propagation equation (7). First, we note that any two classical solutions ψ1(+)(X, Z) and ψ2(+)(X, Z) to Eq. (7) must obey the equations

iΨ1(+)Z=LΨ1(+)2c(Φ0(+))2Ψ1()
iΨ2(+)Z=LΨ2(+)2c(Φ0(+))2Ψ2()
(24)

where the operation L is defined as

L2X2+4cΦ0(+)2.

By combining the first equation (24) with the complex conjugate of the second, the following relation can be deduced by direct computation:

i(Ψ1(+)Ψ2())ZdX=[Ψ2()LΨ1(+)+Ψ1(+)LΨ2()2c(Φ0(+))2Ψ1()Ψ2()
+2c*(Φ0())2Ψ2(+)Ψ1(+)]dX.
(25)

We can simplify this equation to give

iZ(Ψ1(+)Ψ2())dX=[2c(Φ0(+))2Ψ1()Ψ2()
+2c*(Φ0())2Ψ2(+)Ψ1(+)]dX.
(26)

In this form, it is clear that the addition of this equation with its complex conjugate gives zero, that is,

Z[Ψ1(+)(iΨ2())dX+c.c.]=0.
(27)

If we now define the adjoint of ψi(+) by the relation

Ψ̢i(+)(X,Z)=iΨi(+)(X,Z),
(28)

we find that Eq. (27) can be rewritten as

Z[(Ψ2(+)(X,Z)Ψ̢2()(X,Z))dX+c.c.]=0.
(29)

This result shows that in the context of the present problem ψ_ i (+) may be considered to be the adjoint of ψi(+) because it expresses the fact that the inner product of two quantities remains invariant upon propagation. This of course is not the usual definition of the adjoint operation. The adjoint is usually defined within the context of an eigenvalue problem. In such a circumstance, the functions ψ1(+)(X, Z) and ψ2(+)(X, Z) would have the form

Ψi(+)(X,Z)=fi(X)exp(iEiZ)
(30)

where fi (X) is the eigenfunction and Ei is the eigenvalue. If this form is substituted into Eq. (29) we find that the equation is satisfied for E 1E 2 if

f1(X)f2*(X)dX=0,
(31)

which is the usual definition of the adjoint operation.

Appendix B

In this appendix we prove the basic relations (14) which show that the propagation of a spatial soliton can be described in terms of the quantum mechanical equations of motion of a free particle. Through use of the third of Eqs. (13) and the last of the definitions (10) for ψX(+)(X, Z), we find that Δ(Z) can be written as

Δp̂(Z)=1N0Ψ̲X(+)(X,Z)*Ψ̂(+)(X,Z)dX
(32)

We note that the integrals appearing in this equation are of the form of those appearing in Eq. (29) and therefore that

ddZΔp̂(Z)=0.
(33)

Ψ̢p(+)(X,Z)*exp(iN02c2Z4)=Ψ̢p(+)(X,Z)*2ZΨ̢X(+)(X,Z)*
(34)

and thus that Δ(Z) can be written as

ΔX̂(Z)=1N0Ψ̢p(+)(X,Z)*Ψ̂(+)(X,Z)dX+c.c.
+2Z1N0Ψ̢X(+)(X,Z)*Ψ̂(+)(X,Z)dX+c.c..
(35)

Through use of (29) this equation can be rewritten as

ddZΔX̂(Z)=2N0Ψ̢X(+)(X,Z)*Ψ̂(+)(X,Z)dX+H.c.=2Δp̂(Z)
(36)

where we have used the definition (32). Note the key role played by the result (29), which is a property of any two solutions of Eqs. (24).

References and links

1.

A pedagogical discussion of self-action effects including self-trapping and optical solitons is presented in Chapter 6 of R. W. Boyd, Nonlinear Optics (Academic, San Diego, 1992). See also E. M. Nagasako and R. W. Boyd, in Amazing Light, A volume dedicated to Charles Hard Townes on his 80th Birthday, edited by R. Y. Chiao (Springer, New York, 1996).

2.

G. A. Askar’yan, Sov. Phys. JETP15, 1088 (1962).

3.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett.13, 479 (1964). [CrossRef]

4.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP34, 62 (1972).

5.

E. L. Dawes and J. H. Marburger, Phys. Rev.179, 862 (1969); See also J. H. Marburger, Prog. Quantum Electron.4, 35 (1975). [CrossRef]

6.

Y. Silberberg, Opt. Lett.15, 1282 (1990). [CrossRef] [PubMed]

7.

J. S. Aitchison, Y. Silberberg, A. M. Weiner, D. E. Leaird, M. K. Oliver, J. L. Jackel, E. M. Vogel, and P. W. E. SmithJ. Opt. Soc. Am. B8, 1290 (1991). [CrossRef]

8.

M. Shalaby and A. Barthelemy, Opt. Comm.94, 341 (1992). [CrossRef]

9.

A. Villeneuve, J. S. Aitchison, J. U. Kang, P. G. Wigley, and G. I. Stegeman, Opt. Lett.19, 761 (1994). [CrossRef] [PubMed]

10.

B. Luther-Davies and X. Yang, Opt. Lett.17, 1755 (1992). [CrossRef] [PubMed]

11.

S. Blair, K. Wagner, and R. McLeod, Opt. Lett.19, 1943 (1994). [CrossRef] [PubMed]

12.

E. M. Nagasako, R. W. Boyd, and G. S. Agarwal, Phys. Rev. A55, 1412 (1997). [CrossRef]

13.

J. D. Gordon and H. A. Haus, Opt. Lett.11, 665 (1986). [CrossRef] [PubMed]

14.

P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamomoto, Nature, 365, 307 (1993). [CrossRef]

15.

H. A. Haus and M. N. Islam, IEEE J. Quantum Electron.QE-21, 1172 (1985). [CrossRef]

16.

H. A. Haus and Y. Lai, J. Opt. Soc. Am. B7, 386 (1990). [CrossRef]

17.

Although c has the fixed value -1/2 using the present conventions, we retain c in our formulas for more ready comparisons of our results with those obtained using different normalization conventions.

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(270.5530) Quantum optics : Pulse propagation and temporal solitons

ToC Category:
Research Papers

History
Original Manuscript: May 5, 1998
Revised Manuscript: May 5, 1998
Published: August 31, 1997

Citation
Elna Nagasako, Robert Boyd, and Girish Saran Agarwal, "Vacuum-induced jitter in spatial solitons," Opt. Express 3, 171-179 (1998)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-5-171


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References

  1. A pedagogical discussion of self-action effects including self-trapping and optical solitons is presented in Chapter 6 of R. W. Boyd, Nonlinear Optics (Academic, San Diego, 1992). See also E. M. Nagasako, R. W. Boyd, in Amazing Light, A volume dedicated to Charles Hard Townes on his 80th Birthday, edited by R. Y. Chiao (Springer, New York, 1996).
  2. G. A. Askar'yan, Sov. Phys. JETP 15, 1088 (1962).
  3. R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964). [CrossRef]
  4. V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
  5. E. L. Dawes and J. H. Marburger, Phys. Rev. 179, 862 (1969); See also J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975). [CrossRef]
  6. Y. Silberberg, Opt. Lett. 15, 1282 (1990). [CrossRef] [PubMed]
  7. J. S. Aitchison, Y. Silberberg, A. M. Weiner, D. E. Leaird, M. K. Oliver, J. L. Jackel, E. M. Vogel, and P. W. E. Smith, J. Opt. Soc. Am. B 8, 1290 (1991). [CrossRef]
  8. M. Shalaby and A. Barthelemy, Opt. Comm. 94, 341 (1992). [CrossRef]
  9. A. Villeneuve, J. S. Aitchison, J. U. Kang, P. G. Wigley, and G. I. Stegeman, Opt. Lett. 19, 761 (1994). [CrossRef] [PubMed]
  10. B. Luther-Davies and X. Yang, Opt. Lett. 17, 1755 (1992). [CrossRef] [PubMed]
  11. S. Blair, K. Wagner, and R. McLeod, Opt. Lett. 19, 1943 (1994). [CrossRef] [PubMed]
  12. E. M. Nagasako, R. W. Boyd, and G. S. Agarwal, Phys. Rev. A 55, 1412 (1997). [CrossRef]
  13. J. D. Gordon and H. A. Haus, Opt. Lett. 11, 665 (1986). [CrossRef] [PubMed]
  14. P. D. Drummond R. M. Shelby, S. R. Friberg, and Y. Yamomoto, Nature 365, 307 (1993). [CrossRef]
  15. H. A. Haus and M. N. Islam, IEEE J. Quantum Electron. QE-21, 1172 (1985). [CrossRef]
  16. H. A. Haus and Y. Lai, J. Opt. Soc. Am. B 7, 386 (1990). [CrossRef]
  17. Although c has the fixed value -1/2 using the present conventions, we retain c in our formulas for more ready comparisons of our results with those obtained using different normalization conventions.

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